QM, GR Unification from a Third Theory

In a popular book on quantum gravity "Physics meets Philo on the Planck Scale", it is mentioned there are 4 roads to quantum gravity:

"1. quantising General Relativity
2. quantising a different classical theory, while still having general relativity emerge as a low- energy (large-distance) limit.
3. having general relativity emerge as a low-energy limit of a quantum theory that is not a quantization of a classical theory
4. having both general relativity and quantum theory emerge from a theory very different from both"

Can anyone give an example of the fourth road, this part about having both general relativity and quantum theory emerge from a theory very different from both? Is there really such? How many theories along this line? How viable?

String theory is an example of the 4th road or better yet approach. Remember superstring theory quantizes a classical ‘string theory’ and yet has general relativity as a low-energy limit.

The 4th approach is along this line mentioned in book "Physics Meets Philo in the Planck Scale"

For these reasons, a good case can be made that a complete theory of quantum
gravity may require a revision of quantum theory itself in a way that removes the a priori use of continuum numbers in its mathematical formalism.

Finally, we note that (from time to time) a few hardy souls have suggested that a full theory of quantum gravity may require changing the foundations of mathematics itself. A typical argument is that standard mathematics is based on set theory, and certain aspects of the latter (for example, the notion of the continuum) are grounded ultimately in our spatial erceptions.However, our perceptions probe only the world of classical physics – and hence we feed into the mathematical structures currently used in all domains of physics, ideas that are essentially classical in nature. The ensuing category error can be remedied only by thinking quantum theoretically from the very outset – in other words, we must look for ‘quantum analogues’ of the categories of standard mathematics.

How this might be done is by no means obvious.51 One approach is to claim that, since classical logic and set theory are so closely linked (a proposition P determines – and is determined by – the class of all entities for which P can be rightly asserted), one should start instead with the formal structure of quantum logic and try to derive an analogous ‘non-Boolean set theory’. Such ideas are related to the exciting subject of topos theory, which can be viewed as a far-reaching generalization of standard set theory. This is why, as mentioned in Section 2.5.3, topos theory is a natural arena within which to develop speculative schemes in which
‘regions’ of spacetime (or space, or time) are more important than ‘points’ (which may not exist at all).

As a mathematician I strongly object to the notion that mathematics has to change to accommodate physics. If physical theory is inconsistent with a continuum model, then physics needs to look for a new model. The continuum as a mathematical concept will remain.

As stated in the quote maybe some new mathematics needs to be developed, but that doesn't negate the mathematics that is here already.

As a mathematician I strongly object to the notion that mathematics has to change to accommodate physics. If physical theory is inconsistent with a continuum model, then physics needs to look for a new model. The continuum as a mathematical concept will remain.

As stated in the quote maybe some new mathematics needs to be developed, but that doesn't negate the mathematics that is here already.

Do you agree with the following comment by the same author that:

"Despite the variety of programmes, and of controversies, in quantum gravity, most workers would agree on the following, admittedly very general, diagnosis of what is at the root of most of the conceptual problems of quantum gravity. Namely: general relativity is not just a theory of gravity – in an appropriate sense, it is also a theory of spacetime itself; and hence a theory of quantum gravity must have something to say about the quantum nature of space and time."

Meaning all these can be resolved by a theory of quantum gravity.. which is not just about planck scale physics but the nature of space and time itself and how they are connected to matter. What do you think?

The 4th approach is along this line mentioned in book "Physics Meets Philo in the Planck Scale"

How this might be done is by no means obvious.51 One approach is to claim that, since classical logic and set theory are so closely linked (a proposition P determines – and is determined by – the class of all entities for which P can be rightly asserted), one should start instead with the formal structure of quantum logic and try to derive an analogous ‘non-Boolean set theory’.

What do you think?

As I understand it quantum logic denies the distribution of disjunction. However, conjunctions and distunctions and all the associative and distrributive laws can be constructed only using implication, negation, and parenthesis. I don't see how it would be possible to affirm these logical connectives on the one hand and deny them in some special case. For me that forms a contradiction.

"Despite the variety of programmes, and of controversies, in quantum gravity, most workers would agree on the following, admittedly very general, diagnosis of what is at the root of most of the conceptual problems of quantum gravity. Namely: general relativity is not just a theory of gravity – in an appropriate sense, it is also a theory of spacetime itself; and hence a theory of quantum gravity must have something to say about the quantum nature of space and time."

Meaning all these can be resolved by a theory of quantum gravity.. which is not just about planck scale physics but the nature of space and time itself and how they are connected to matter. What do you think?

It seems that general relativity should be derivable from some quantum theory since quantum theory is the theory of the small and general relativity is a theory of the huge. I would think you build larger structures from smaller ones.

All quantum theory is built on some sort of spacetime background, the derivatives wrt to time and space. What is needed, in my opinion, is to find the background independent structure behind quantum theory and how to construct a diffeomorphic invariant form of quantum theory. Then if we can construct QT with the metric explicit in the equations, we might then derive the metric dependent quantities in terms of the energy-momentum terms.

As I understand it quantum logic denies the distribution of disjunction. However, conjunctions and distunctions and all the associative and distrributive laws can be constructed only using implication, negation, and parenthesis. I don't see how it would be possible to affirm these logical connectives on the one hand and deny them in some special case. For me that forms a contradiction.

It seems that general relativity should be derivable from some quantum theory since quantum theory is the theory of the small and general relativity is a theory of the huge. I would think you build larger structures from smaller ones.

All quantum theory is built on some sort of spacetime background, the derivatives wrt to time and space. What is needed, in my opinion, is to find the background independent structure behind quantum theory and how to construct a diffeomorphic invariant form of quantum theory. Then if we can construct QT with the metric explicit in the equations, we might then derive the metric dependent quantities in terms of the energy-momentum terms.

Our QFT works on a fixed background where particles move against a steady background. We must develope theory where QFT and spacetime are one interacting thing (background independence). Any good candidates for it?

Our QFT works on a fixed background where particles move against a steady background. We must develope theory where QFT and spacetime are one interacting thing (background independence). Any good candidates for it?

Can we start with the assumption that General Relativity and Quantum Mechanics are valid in all detail at every level of reality and find out what the common basis of both are? I'm not sure this bottom-up approach will work. We don't even know if they are valid at every level.

Or maybe we can start from some principle from which to develop both GR and QM. But what principle would that be? I suppose you'd need to be able to completely justify whatever starting principle you used in this top-down approach.

Loop Quantum Gravity seems to be a bottom-up approach that assume GR and QM is valid to start with. Superstring theory seems to be a top-down approach whose first principle is that all interactions can be describe by vibrating strings. Maybe we can categorize other approaches as to whether they are bottom-up or top-down.

I'm not satisfied with either. I'm not comfortable trying to derive things by assuming they are correct as in Loop Quantum Gravity. And it seems the vibrating strings of M-theory is not well motivated. Neither appoach seems to have an explantion for where quantum theroy comes from to begin with; they simply apply it to their efforts as needed.

The bottom-up approach would seem circular when trying to find the most fundamental appoach. But then what would be the completely justifiable basic principle used in a top-down approach? The only way to completely justify physics, so that it would not be possible to even ask any more question, would be to derive physics from logic itself. That seems like a daunting task, but perhaps we are closer than we think.

Can we start with the assumption that General Relativity and Quantum Mechanics are valid in all detail at every level of reality and find out what the common basis of both are? I'm not sure this bottom-up approach will work. We don't even know if they are valid at every level.

Or maybe we can start from some principle from which to develop both GR and QM. But what principle would that be? I suppose you'd need to be able to completely justify whatever starting principle you used in this top-down approach.

Loop Quantum Gravity seems to be a bottom-up approach that assume GR and QM is valid to start with. Superstring theory seems to be a top-down approach whose first principle is that all interactions can be describe by vibrating strings. Maybe we can categorize other approaches as to whether they are bottom-up or top-down.

I'm not satisfied with either. I'm not comfortable trying to derive things by assuming they are correct as in Loop Quantum Gravity. And it seems the vibrating strings of M-theory is not well motivated. Neither appoach seems to have an explantion for where quantum theroy comes from to begin with; they simply apply it to their efforts as needed.

The bottom-up approach would seem circular when trying to find the most fundamental appoach. But then what would be the completely justifiable basic principle used in a top-down approach? The only way to completely justify physics, so that it would not be possible to even ask any more question, would be to derive physics from logic itself. That seems like a daunting task, but perhaps we are closer than we think.

Happy New Year.

Is there any counterparts of Copenhagen, Many Worlds and Bohmian Mechanics in Spacetime? We can't tell by observation whether electrons inhabit space-time as point-like particles, or distributed mass fields, or intermittent point events or just in the equations. Likewise, we don't know whether spacetime structure are really there or just in the equations. Maybe all the spacetime theories we have are like Copenhagen in that we only deal with the equations and not something physical like Many Worlds or Bohmian Mechanics or are there also versions of these two in spacetime? I think Loop Quantum Gravity and Super string theory are like Copenhagen. In LQG, the spin networks are just in the mathematics and not really in physical space.

Is there any counterparts of Copenhagen, Many Worlds and Bohmian Mechanics in Spacetime? We can't tell by observation whether electrons inhabit space-time as point-like particles, or distributed mass fields, or intermittent point events or just in the equations. Likewise, we don't know whether spacetime structure are really there or just in the equations. Maybe all the spacetime theories we have are like Copenhagen in that we only deal with the equations and not something physical like Many Worlds or Bohmian Mechanics or are there also versions of these two in spacetime? I think Loop Quantum Gravity and Super string theory are like Copenhagen. In LQG, the spin networks are just in the mathematics and not really in physical space.

Happy New Year.

I think the answer to these questions require an understanding of the foundations of quantum mechanics and where it can be used. So I think quantum gravity will not be solved until we understand why nature is quantum mechanical in the first place.

What are the big issues with quantising gravity? Is it correct that one issue is that at the quantum level spacetime is chaotic?

As I understand it, we're not even sure if it is legitimate to quantize the gravitational field - much less how. We seem to be assuming that just because it is a field that it can be quantized. But it is different because all the other fields that are quantized, are quantized with respect to the spacetime coordinates that make up the metric. So it's not clear that you should quantize the metric.

As I understand it, we're not even sure if it is legitimate to quantize the gravitational field - much less how. We seem to be assuming that just because it is a field that it can be quantized. But it is different because all the other fields that are quantized, are quantized with respect to the spacetime coordinates that make up the metric. So it's not clear that you should quantize the metric.

If there is a minimum amount of energy that can used, then does it not follow that there is a minimum amount of distance that an object can be moved by that energy and a minimum on the amount of time that it can take to do it? This sounds like quantisation.